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3.2 Three Ways to Prove Triangles Congruent Objectives: 1. identify included angles and included sides 2. apply the SSS postulate 3. apply the SAS postulate 4. apply the ASA postulate Triangles Use a ruler and protractor to construct a triangle with the following specifications: a. One side is 2cm b. One side is 3cm c. One side is 4cm Side-Side-Side (SSS) Postulate If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. Triangles Use a ruler and protractor to construct a triangle with the following specifications: a. One side is 5cm b. One side is 8cm c. One angle is 40Λ Side-Angle-Side (SAS) Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Triangles Use a ruler and protractor to construct a triangle with the following specifications: a. One side is 9cm b. One angle is 50Λ c. One angle is 30Λ Angle-Side-Angle (ASA) Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Three Ways to Prove Triangles Congruent 1. SSS Postulate 2. SAS Postulate 3. ASA Postulate Cannot Use SSA or AAA Example Name the congruent angles or sides necessary to prove the triangles congruent by the specified method. a. SSS b. SAS Example Name the congruent angles or sides necessary to prove the triangles congruent by the specified method. a. SAS b. ASA Example Name the congruent angles or sides necessary to prove the triangles congruent by the specified method. a. SAS b. ASA Example Name the postulate (if any) that can be used to prove the triangles congruent. Proof Given: Diagram Prove: βπΊπ»π½ β βπΊπΎπ½ SSS Postulate or SAS Postulate or ASA Postulate Proof Given: Diagram Prove: βπΊπ»π½ β βπΊπΎπ½ Statements 1. 2. 3. 4. πΊπ» β πΊπΎ π»π½ β πΎπ½ πΊπ½ β πΊπ½ βπΊπ»π½ β βπΊπΎπ½ Reasons 1. 2. 3. 4. Given (indicated in diagram) Given (indicated in diagram) Reflexive Property SSS Postulate Flow Proof Given: Diagram Prove: βπΊπ»π½ β βπΊπΎπ½ Proof Given: π΄π· β π΄πΈ π΄π΅ β π΄πΆ Prove: βπ΄π·π΅ β βπ΄πΈπΆ SSS Postulate or SAS Postulate or ASA Postulate Proof Given: π΄π· β π΄πΈ π΄π΅ β π΄πΆ Prove: βπ΄π·π΅ β βπ΄πΈπΆ Statements 1. π΄π· β π΄πΈ 2. π΄π΅ β π΄πΆ 3. β π΄ β β π΄ 4. βπ΄π·π΅ β βπ΄πΈπΆ Reasons 1. Given 2. Given 3. Reflexive Property 4. SAS Postulate Assignment Section 3.2 Problem Set A #1-10 Proof Given: π΄π· β πΆπ· B is the midpoint of π΄πΆ Prove: βπ΄π΅π· β βπΆπ΅π· SSS Postulate or SAS Postulate or ASA Postulate Proof Given: π΄π· β πΆπ· B is the midpoint of π΄πΆ Prove: βπ΄π΅π· β βπΆπ΅π· Statements Reasons 4. π·π΅ β π·π΅ 5. βπ΄π΅π· β βπΆπ΅π· 1. Given 2. Given 3. If a point is a midpoint, then it divides the segment into two congruent segments. 4. Reflexive Property 5. SSS Postulate 1. π΄π· β πΆπ· 2. B is the midpoint of π΄πΆ 3. π΄π΅ β πΆπ΅ Proof Given: β 3 β β 6 πΎπ β ππ β πΎπ π β β ππ π Prove: βπΎπ π β βππ π SSS Postulate or SAS Postulate or ASA Postulate Proof Given: β 3 β β 6 πΎπ β ππ β πΎπ π β β ππ π Prove: βπΎπ π β βππ π Statements Reasons 1. β 3 β β 6 2. β πΎππ is straight 3. β 4 is supp. to β 3 4. β 5 is supp. to β 6 5. β 4 β β 5 6. πΎπ β ππ 7. β πΎπ π β β ππ π 8. β πΎπ π β β ππ π 9. βπΎπ π β βππ π 1. 2. 3. 4. 5. 6. 7. 8. 9. Given Given Given ASA Postulate Proof Given: β 3 β β 6 πΎπ β ππ β πΎπ π β β ππ π Prove: βπΎπ π β βππ π Statements Reasons 1. β 3 β β 6 2. β πΎππ is straight 3. β 4 is supp. to β 3 4. β 5 is supp. to β 6 5. β 4 β β 5 6. πΎπ β ππ 7. β πΎπ π β β ππ π 8. β πΎπ π β β ππ π 9. βπΎπ π β βππ π 1. Given 2. Assumed from diagram 3. If the sum of two angles is a straight angle, then the angles are supplementary. 4. If the sum of two angles is a straight angle, then the angles are supplementary. 5. If two angles are supplementary to congruent angles, then they are congruent. 6. Given 7. Given 8. If an angle is subtracted from two congruent angles, then the differences are congruent. 9. ASA Postulate Assignment Section 3.2 Problem Set A #12-16 Problem Set B #17